Theorem.If converges then tends to as approaches in such a way that remains bounded.

Here was assuming that and convergence takes place at .

But we let and for example, right?

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- March 26th 2009, 10:46 PMmanjohn12Abel's Limit Theorem
**Theorem.**If converges then tends to as approaches in such a way that remains bounded.

Here was assuming that and convergence takes place at .

But we let and for example, right? - March 29th 2009, 11:52 AMOpalg
That is correct. The usual statement of Abel's theorem is that if the series has radius of convergence R, and converges at a point on the circle of convergence (so ), then as

*nontangentially*(meaning that that the angle between and the tangent at is bounded away from 0).